in which \(f=-\xi ^{\mu }\xi _{\mu }\)  and \(D_{i}\)  is the spatial covariant
derivative on \(\Sigma \) . We assume that the Hilbert space \(\mathcal {H}=L^{2}\left( \Sigma , \mu \right)\)  is the space of square integrable
functions on \(\Sigma \)  with appropriate measure \(\mu \) . Initially the operator \(A\)  is defined on smooth functions with compact support \(C_{0}^{\infty }(\Sigma )\) . Since the operator \(A\)  is real,
positive and symmetric its self-adjoint extensions always exist.
If it has a unique extension \(A_{E},\)  then \(A\)  is called essentially
self-adjoint [1]}, [2]}, [3]}. Accordingly, the Klein-Gordon equation for
a free particle satisfies
\(i\frac{d\psi }{dt}=\sqrt{A_{E}}\psi ,\) 
